Nonlinear Thermally Induced Vibration Analysis of Porous FGM Timoshenko Beams Embedded in an Elastic Medium

In this paper, the geometrically nonlinear thermally induced vibration response of beams made of porous functionally graded materials (FGMs) under thermal shock is investigated using a novel numerical approach. The material properties are considered to be temperature- and position-dependent. It is also assumed that the beams are embedded in an elastic medium which is considered as Winkler-Pasternak type. Hamilton's principle, the Timoshenko beam theory and the von Karman geometrical nonlinear assumptions are used to derive the equations of motion. The matrix representation of relations is given that can be efficiently utilized in numerical approaches. Solution in time domain is done using the Newmark algorithm according to the constant average acceleration technique. In the space domain, the generalized differential quadrature and variational differential quadrature (VDQ) methods are employed for discretization. Using VDQ leads to compact/efficient vector-matrix relations which can be readily utilized in the coding process. The numerical results are presented to analyze the effects of power law index, boundary conditions, material porosity, elastic foundation and length-to-thickness ratio on the nonlinear thermally induced vibrations of FGM porous beams. It is shown that considering the even distribution, the vibration amplitude of beams decreases as the porosity volume fraction gets larger, while it increases for the uneven distribution. Moreover, the time required to reach the steady state of vibrations increases with increasing the power law index of FGM.